RESOURCE LETTER

RESOURCE LETTER
Resource Letters are guides for college and university physicists, astronomers, and other scientists to literature, websites, and other teaching aids.
Each Resource Letter focuses on a particular topic and is intended to help teachers improve course content in a specific field of physics or to
introduce nonspecialists to this field. The Resource Letters Editorial Board meets at the AAPT Winter Meeting to choose topics for which Resource
Letters will be commissioned during the ensuing year. Items in the Resource Letter below are labeled with the letter E to indicate elementary level
or material of general interest to persons seeking to become informed in the field, the letter I to indicate intermediate level or somewhat specialized
material, or the letter A to indicate advanced or specialized material. No Resource Letter is meant to be exhaustive and complete; in time there may
be more than one Resource Letter on a given subject. A complete list by field of all Resource Letters published to date is at the website
www.kzoo.edu/ajp/letters.html. Suggestions for future Resource Letters, including those of high pedagogical value, are welcome and should be sent
to Professor Roger H. Stuewer, Editor, AAPT Resource Letters, School of Physics and Astronomy, University of Minnesota, 116 Church Street SE,
Minneapolis, MN 55455; e-mail: rstuewer@physics.umn.edu
Resource Letter BH-2: Black Holes
Elena Galloa兲
Massachusetts Institute of Technology, Kavli Institute for Astrophysics and Space Research,
70 Vassar Street, Building 37-685, Camdridge, Massachusetts 02139
Donald Marolfc兲
Department of Physics, University of California Santa Barbara, Santa Barbara, California 93106-9530
共Received 4 June 2008; accepted 5 December 2008兲
This Resource Letter is designed to guide students, educators, and researchers through 共some of兲 the
literature on black holes. We discuss both the physics and astrophysics of black holes. We emphasize
breadth over depth, and review articles over primary sources. We include resources ranging from
nontechnical discussions appropriate for broad audiences to technical reviews of current research.
Topics addressed include classification of stationary solutions, perturbations and stability of black
holes, numerical simulations, collisions, the production of gravity waves, black-hole
thermodynamics and Hawking radiation, quantum treatments of black holes, black holes in both
higher and lower dimensions, and connections to nuclear and condensed-matter physics. On the
astronomical end, we also cover the physics of gas accretion onto black holes, relativistic jets,
gravitationally redshifted emission lines, evidence for stellar-mass black holes in binary systems and
supermassive black holes at the centers of galaxies, the quest for intermediate-mass black holes, the
assembly and merging history of supermassive black holes through cosmic time, and their affects on
the evolution of galaxies. © 2009 American Association of Physics Teachers.
关DOI: 10.1119/1.3056569兴
I. INTRODUCTION
The status of black-hole physics has changed dramatically
since the publication of the first black-hole Resource Letter,
Ref. below. The observational evidence for black holes is
now plentiful and strong. Moreover, black holes have progressed from a curiosity of mathematical physics to a tool for
building or studying models in other branches of physics.
Examples range from feedback mechanisms that may control
the formation and evolution of galaxies 共see Refs. 66 and 67兲
to string-theoretic connections to nuclear and condensed
matter physics 共see Refs. 36, 151, and 152兲. Nevertheless,
black holes remain fascinating objects in their own right that
may yet divulge deep clues to further revolutions in fundamental physics.
We have endeavored to provide a guide to this extraordinarily diverse literature, and to include resources at a variety
of levels. Some advanced resources will require additional
specialized knowledge of general relativity, astrophysics, or
a兲
Electronic mail: egallo@mit.edu
c兲
Electronic mail: marolf@physics.ucsb.edu
294
Am. J. Phys. 77 共4兲, April 2009
http://aapt.org/ajp
quantum-field theory. We hope that bringing together many
levels of material will allow students of black-hole physics to
find the right entry point for their background and teachers of
black-hole physics to extract useful material for their
courses.
Given the breadth of our subject, it is impossible to be
complete. We have by no means attempted to do so. Experts
will find many of their favorite resources, and perhaps even
their favorite topics to be missing entirely. We have concentrated on a “few” review articles and seminal papers, though
we have tried to include sufficient commentary to help the
user locate the most useful resource. Where possible, we
have chosen review articles over original works and modern
treatments over classic presentations. We made this decision
for purely practical reasons, and we encourage the reader to
consult Steve Detweiler’s original black-hole resource letter
共Ref. below兲 for an excellent guide to the historical literature.
Indeed, we view our work as a supplement to his, not as a
replacement.
1. “Resource Letter BH-1: Black Holes,” S. Detweiler,
Am. J. Phys. 49, 394–400 共1981兲.
Resources in the following sections are organized first by
© 2009 American Association of Physics Teachers
294
level: popular material appears in Sec. II while most U.S.
undergraduate material appears in Sec. III. Graduate generalrelativity textbooks are found in Sec. IV while more specialized material is presented in Secs. V–VIII. However, some of
the material in these latter sections is also of intermediate
level, accessible to advanced undergraduate students. We
have then attempted to sort the more specialized papers by
topic. This task is far from straightforward, as many excellent reviews overlap several obvious categories. The result is
that many resources are cross-referenced, typically by adding
a note to “see also Ref. nnn.” The reader is well-advised to
do so.
We have arranged the specialized material as follows: Section V contains resources dealing with black holes in astrophysics. This includes observations of black holes, models of
astrophysical processes involving black holes, and the implications of these models.
Section VI then addresses well-established mathematical
physics describing the statics, dynamics, and thermodynamics of black holes. The focus here is on classical black-hole
physics such as the treatment and classification of black
holes, in general relativity and related theories 共in a variety
of dimensions兲, their perturbations and stability properties,
their formation and interaction through collisions, the production of gravitational radiation and the relation to gravitywave detectors, and black-hole thermodynamics. In addition,
however, Sec. VI D includes broadly accepted material on
quantum field theory in curved spacetime and the associated
Hawking radiation. Such subjects are deeply connected to
the thermodynamics of black holes and are addressed in
much of the same literature. Discussions of the possible production of black holes in particle collisions has also been
included in Sec. VI E.
In contrast, those references that concern the deep quantum microphysics of black holes have been placed in Sec.
VII. Such resources are more speculative, at least in the
sense that they typically make assumptions about the quantum nature of gravity that are beyond the reach of current
experiments. We therefore separate this material from that of
Sec. VI D, even though Hawking radiation is again a common theme. Finally, we have collected connections to
nuclear and condensed-matter physics in the short Sec. VIII.
Such connections include the use of established fluid dynamics and condensed-matter physics to describe certain analogues of black holes, as well as current attempts to use
black-hole physics 共and string theory兲 to describe as yet
poorly understood effects in both nuclear and condensedmatter physics. Here the emphasis is again on classical
black-hole physics but in novel, and sometimes somewhat
speculative contexts.
A. Basic resources and notation
Perhaps the most important information we can convey is
how the reader might find and access useful works that will
appear only after the publication of this Resource Letter.
Web-based search engines continue to make this more and
more possible, but it is worthwhile to point out a few places
to start. When entering a new subject, it is natural to begin
with a review article. Several journals are dedicated entirely
to such reviews, and one may begin by searching their websites. Certain online tools are also of particular help in locating reviews. Some suggestions are:
295
Am. J. Phys., Vol. 77, No. 4, April 2009
共1兲 Living Reviews in Relativity.
This online journal publishes only review articles on gravitational physics.
As you might guess, a large fraction of its contents deal with black holes,
making it a prime resource. Indeed, many of the articles described below are
from this journal.
共2兲 Reviews of Modern Physics.
A review journal published by the American Physical Society. Topics covered span all of physics.
共3兲 Physics Reports.
A review journal by Elsevier. Topics covered span all of physics.
共4兲 HEP Reviews at SPIRES, www.slac.stanford.edu/spires/reviews/.
An ongoing project to index online reviews in high-energy and theoretical
physics. Black holes are currently category Id2. This website also explains
how to perform a more general search of the SPIRES database for relevant
review articles.
共5兲 The SAO/NASA Astrophysics Data System Abstract Service 共ADS兲,
adsabs.harvard.edu/abstractគservice.html/.
Hosted by the Smithsonian Astrophysical Observatory for NASA, this website is the most powerful browser for astrophysics-related material. It allows
to search for, e.g., specific authors 共or list thereof兲, refereed vs nonrefereed
publications, works mentioning your favorite astronomical object, as well as
title keywords, publication year, number of citations, etc. It covers a wide
spectrum of levels, from scholarly research papers, to Ph.D. thesis, to conference proceedings. A full description of the search criteria is given at:
doc.adsabs.harvard.edu/absគdoc/helpគpages/
basicsearch.html#BasicគQueryគForm. ADS has become an irreplaceable instrument in the workday of every astrophysicist.
共6兲 Annual Reviews of Astronomy and Astrophysics,
arjournals.annualreviews.org/loi/astro/.
ARAA publishes comprehensive reviews written by well-known authors.
They are an excellent starting point for the interested reader who wishes to
approach a new astrophysical subject at the intermediate-advanced level.
The resources that we list below should not be hard to
locate. Most appear in journals that can be found in a typical
university library or online. In addition, most also include an
“Arxiv identifier” either of the form 关arXiv:astro-ph/
yymmnnn兴 or 关arXiv:yymm.nnnn 关astro-ph兴兴, with astroph
perhaps replaced by qr-qc, hep-th, or hep-ph. These reference
numbers refer to the online physics archive currently maintained by the Cornell University Library: arxiv.org/. Many
papers can be found here by using the search feature. To find
the precise desired work, simply search by “identifier” and
enter it into the search box: e.g., astroph/yymmnnn in the
first 共older style兲 numbering system or simply yymm.nnnn in
the second system. Here yy denotes the year, mm the month,
and nnn or nnnn the number assigned to the work when it
was uploaded to arxiv.org.
As a final remark, the reader should be aware that two
acronyms are frequently used in the text below: General relativity is abbreviated “GR,” and “AGN” refers to active galactic nuclei. We have otherwise attempted to make the comments following each resource as approachable as possible
by the general reader.
II. POPULAR MATERIAL
Material in this section should be accessible to persons
with a high school or college-level education without specialized training in physics. The goal of these works is typically to convey the general idea of the subject, without going
into technical details. Certain resources are nevertheless
somewhat demanding of the reader.
A. Popular books
2. Black Holes and Time Warps: Einstein’s Outrageous
Legacy, K. S. Thorne 共W.W. Norton, New York, 1994兲. AlE. Gallo and D. Marolf
295
though it is now 14 years old, this book remains the most
popular one on this subject. First and foremost, it is an accessible introduction to basic issues: What is a black hole?
How are they made? What happens when you fall in? The
text begins by explaining the essential ideas in relativity and
proceeds through the development of black-hole physics itself. The approach is historical, and the author’s personal
involvement in this history cements a gripping story line. But
it is the style of the text that captivates many readers. The
author’s charming personality shines through. The final
chapter on wormholes should be regarded as more speculative material, but the rest of the book focuses on wellestablished physics. 共E兲
3. Black Hole Physics: Basic Concepts and New Developments, V. P. Frolov and I. D. Novikov 共Springer, New
York, 1998兲. An encyclopedic, self-contained text on the
physics of black holes, with a special focus on black holes in
astrophysics 共I兲.
4. Gravity’s Fatal Attraction: Black Holes in the Universe, M. Begelman and M. Rees 共Scientific American Library, New York, 1995兲. Do “spacetime singularities” actually exist in nature? Have we found them? Begelman and
Rees tell us the captivating story of the scientific
endeavors—and related anecdotes—that led to the discovery
of black holes. Powerful, inspiring, and fun to read. 共E兲
5. General Relativity from A to B, R. Geroch 共U. of
Chicago Press, Chicago, 1978兲. Developed as lectures to
nonscience majors at the University of Chicago, Geroch’s
book gives a solid conceptual explanation of special and GR.
Some math is used. This book requires more effort than those
above, but delivers a deeper understanding of relativity. 共E/I兲
6. Was Einstein Right? Putting General Relativity to
the Test, C. M. Will 共BasicBooks, New York, 1993兲. Most
popular GR books emphasize the theory and conceptual underpinnings of the theory. In contrast, Will’s book focuses on
the many empirical tests of this theory, including Hulse and
Taylor’s Nobel-prize winning verification of the decay of
neutron star orbits via gravitational radiation. 共E/I兲
7. Relativity, A. Einstein, 共Three Rivers, New York,
1961兲. Most later expositions of GR are based at least in part
on this book, Einstein’s attempt to describe GR to anyone
with a standard education. In his own words, “despite the
shortness of the book, a fair amount of patience and force of
will” is required of the reader. Nevertheless, the book is an
invaluable opportunity to learn at least a bit of relativity from
Einstein himself. 共I兲
B. Websites
We highly recommend the websites below, all at the elementary level. Of course, websites are in constant flux.
Check your favorite search engine.
8. en.wikipedia.org/wiki/Blackគhole/
Black Hole, Wikipedia. As usual, this Wikipedia entry is surprisingly broad. The reference list is excellent. 共E兲.
9. hubblesite.org/exploreគastronomy/blackគholes/
Black Holes: Gravity’s Relentless Pull, Hubble Site. Packed
with graphics, simulated “experiments,” and animations, this
professional site from NASA and the Space Telescope Science Institute provides an exciting tour of black-hole physics. 共E兲
10. www.gothosenterprises.com/blackគholes/
Jillian’s Guide to Black Holes, J. Bornak. A charming, informal, and plain-speaking introduction to black holes and GR.
296
Am. J. Phys., Vol. 77, No. 4, April 2009
Written by a black-hole fan as an undergraduate project.
Nonprofessional in a delightful way and a marvelous introduction to the basic physics. 共E兲
11. www.space.com/blackholes/
All About Black Holes. Lovely videos and simulations, as
well as black-hole news items. 共E兲
12. antwrp.gsfc.nasa.gov/htmltest/rjnគbht.html/
Virtual Trips to Black Holes and Neutron Stars, R. Nemiroff.
The way black holes curve spacetime makes light do crazy
things. What would you see if you approached a black hole?
Check this classic website to find out 共E兲
13. casa.colorado.edu/~ajsh/schw.shtml/
Falling Into a Black Hole, A. Hamilton. More modern simulations of trips to and inside black holes, but with a somewhat confusing portrayal of the horizon. 共E兲
14. www-xray.ast.cam.ac.uk/xray-introduction/Blackhole
binary.html/
Black Holes and X-Ray Binaries. Black holes and astrophysics from the Institute of Astronomy, Cambridge. 共E兲
15. library.thinkquest.org/25715/index.htm/
Event Horizon. Black-hole basics with great simulations. 共E兲
16. www.astro.umd.edu/chris/Research/X-raysគandគBlackគ
holes/x⫺raysគandគblackគholes.html/
X-Rays and Black Holes, C. Reynolds, University of Maryland. A brief, informal and yet accurate description of the
status of research on astrophysical black holes, explained for
the nonexperts by a forefront leader in the field. 共E兲
17. www.youtube.com/watch?v⫽hoLvOvGW3Tk
Black Holes, Neutron Stars, White Dwarfs, Space and Time,
YouTube. Simulations of black holes feeding and being born
through collisions, all set to an exciting sound track. 共E兲
18. amazing-space.stsci.edu/resources/explorations/black
holes/
Amazing Space, Space Telescope Science Institute. This
website provides black-hole resources for teachers and students. 共E兲
C. Popular articles
The reader can find a number of interesting black-hole
articles in popular-science publications such as Scientific
American, Physics World, Physics Today, New Scientist and,
at a slightly more advanced level, Science and Nature. Some
fairly recent articles are listed below, but there is no doubt
that more will appear in the near future. We have sorted these
articles roughly into those that emphasize the astrophysics of
black holes and those emphasizing fundamental aspects of
theoretical physics.
Astrophysics:
19. “Black hole blowback,” W. Tucker, H. Tananbaum, A.
Fabian, Sci. Am. 296, 42–49 共2007兲. Black holes and active
galactic nuclei as possible drivers of feedback in galaxy cluster evolution. 共E兲
20. “Unmasking black holes,” J. Lasota, Sci. Am. 280,
30–37 共1999兲. On how we might directly detect black-hole
horizons. 共E兲
21. “The galactic odd couple,” K. Weaver, Sci. Am. 289,
34–41 共2003兲. Is there a connection between black holes and
bursts of start formation? 共E兲
22. “The midlife crisis of the cosmos,” A. J. Barger, Sci.
Am. 292 32–39 共2005兲. The authors describe the present and
past of star and black-hole formation in our universe. 共E兲
23. “The brightest explosions in the universe,” N. Gehrels,
L. Piro, and P. J. T. Leonard, Sci. Am., 287, 84–91 共2002兲.
E. Gallo and D. Marolf
296
Connecting the birth of black holes to enormous explosions
called gamma-ray bursts. 共E兲
Fundamental physics:
24. “Quantum black holes,” B. J. Carr and S. B. Giddings,
Sci. Am. 292, 48–55 共2006兲. Might we soon be able to make
black holes in the laboratory? 共E兲
25. “The reluctant father of black holes,” J. Bernstein, Sci.
Am., 274, 80–85 共1996兲. The early history of our understanding of black holes and of Einstein’s resistance to accepting them. 共E兲
26. “An echo of black holes,” T. A. Jacobson and R.
Parentani, Sci. Am. 293 68–75 共2006兲. Flowing fluids in the
laboratory can act much like black holes, and even have a
form of Hawking radiation. 共E兲
27. “Black hole computers,” S. Lloyd and Y. J. Ng, Sci.
Am. 291, 52–61 共2004兲. Black Holes and Information
Theory. 共E兲
28. “Information in the holographic universe,” J. D. Bekenstein, Sci. Am. 289, 58–65 共2003兲. How black holes may
be related to fundamental bounds on the information that
objects can hold. 共E兲
29. “The illusion of gravity,” J. Maldacena, Sci. Am. 293,
57–63 共2005兲. String theory has holographic properties that
seem deeply linked to black hole physics. 共E兲
30. “Reality-bending black holes,” Sci. Am. Special Edition, March 2007. This volume collects Refs. 19–29 above,
along with other articles on related topics. 共E兲
gives a broad overview of loop quantum gravity for the nonexpert. After introducing the basic ideas, Sec. III B addresses
black-hole entropy in loop quantum gravity. The article
should be accessible to students familiar with undergraduate
quantum mechanics, the description of electromagnetism via
4-vector potentials, and some knowledge of GR. 共I兲
34. “Black holes and quark confinement,” E. Witten, Curr.
Sci. 81, 1576–1581 共2001兲. Witten takes the reader on a tour
of ideas ranging from strings to quarks to black holes and the
AdS/CFT duality. The climax is the fusion of these ideas into
a stringy derivation of quark confinement in theories mathematically similar to, but still different than the QCD that
describes the strong interactions of our universe. 共I兲
35. “A microscopic theory of black holes in string theory,”
S. R. Wadia, Curr. Sci. 81共12兲, 1591–1597 共2001兲. After providing an undergraduate-level description of black-hole thermodynamics and Hawking radiation, Wadia describes the extent to which these phenomena are currently described by
string theory. The concluding section includes a useful set of
open questions. 共I兲
36. “Theoretical physics: A black hole full of answers,” J.
Zaanen, Nature 共London兲 448, 1000–1001 共2007兲. Zaanen
briefly describes current attempts to use black holes and
string theory as mathematical tools to explain the physics of
both high-temperature superconductors and quark-gluon
plasmas. 共I兲
B. General relativity textbooks
III. UNDERGRADUATE MATERIAL
This section describes resources at the advanced 共U.S.兲
undergraduate level. Some entries should be accessible to
any student who has completed a 1-year calculus-based
physics course, while others require background in intermediate classical mechanics, quantum mechanics, and electromagnetism. The articles in Sec. III A address special topics
not covered by standard textbooks, but which are nevertheless approachable by readers without detailed knowledge of
GR. Students who wish to establish a working understanding
of black holes should consult one of the undergraduate textbooks in Sec. III B.
A. Articles
31. “Black holes: The inside story,” S. Droz, W. Israel and
S. M. Morsink, Phys. World, 9, 34–37 共1996兲. One of a very
few articles with a good description of black-hole interiors.
A key point is the instability of the so-called inner horizon of
charged and rotating black holes. This instability forbids the
use of black holes as passageways to other universes. The
article is also posted online at www.phys.ualberta.ca/
⬃morsink/blackhole/ in both English and French translations. 共E兲
32. “Spacetime embedding diagrams for black holes,” D.
Marolf, Gen. Relativ. Gravit. 31, 919–944 共1999兲; e-print
arXiv:gr-qc/9806123. The curved spacetime of a Schwarzschild black hole can be visualized by drawing the so-called
rt-plane as a curved surface inside flat 2 + 1 Minkowski
space. The main text is accessible to those with some training in special relativity, though some knowledge of GR is
required to follow the construction 共in the appendix兲 of the
diagrams from the black hole metric. 共I兲
33 “Gravity and the quantum,” A. Ashtekar, New J. Phys.
7, 198–198 共2005兲; e-print arXiv:gr-qc/0410054. Ashtekar
297
Am. J. Phys., Vol. 77, No. 4, April 2009
A number of excellent GR textbooks are now available at
the U.S. undergraduate level, each with a good treatment of
black-hole physics. The texts differ in terms of both perspective and level of sophistication, but each one provides what
is in principle a self-contained introduction. We have ordered
the books below roughly in terms of level of presentation.
37. Gravity From the Ground Up, B. Schutz 共Cambridge
U. P., Cambridge, 2003兲. This excellent book emphasizes
gravity, and its interactions with astrophysics. Assuming only
high-school math and physics, Schutz addresses the solar
system, the birth and life cycle of stars, relativity, cosmology,
gravity waves, and gravitational lenses. The chapter on black
holes summarizes a wide range of physics from accretion
and black hole growth to Hawking radiation and black-hole
entropy. 共E/I兲
38. Notes on Relativity and Cosmology, D. Marolf
共2003兲. Available on-line at www.physics.ucsb.edu/~marolf/
MasterNotes.pdf. Designed to convey understanding of relativity using only elementary calculus, these notes first thoroughly treat accelerated reference frames in special relativity.
Spacetime diagrams are favored over calculations. GR is introduced by piecing together such frames using the equivalence principle. The chapter on black holes emphasizes similarities between horizons and accelerated frames in flat
spacetime, and also summarizes Ref. 32 above. 共E/I兲
39. Exploring Black Holes: Introduction to General
Relativity, E. F. Taylor and J. A. Wheeler 共Addison Wesley
Longman, San Francisco, 2000兲. Taylor and Wheeler focus
on a nuts-and-bolts exploration of black-hole solutions as an
introduction to GR. The text requires algebra and basic calculus but does not assume prior knowledge of physics or
relativity. The emphasis is on the experience of an observer
near a black hole. 共I兲
40. Flat and Curved Space-Times, 2nd ed. by G. F. R.
Ellis and R. Williams 共Oxford U. P., New York, 2000兲. This
E. Gallo and D. Marolf
297
excellent book on special and general relativity includes a
survey of basic black-hole results including no-hair theorems, energy extraction, and thermodynamics. Ellis and Williams assume knowledge of basic calculus and introduce
other mathematics as needed. 共I兲
41. Gravity, An Introduction to Einstein’s General
Relativity, J. B. Hartle 共Addison Wesley, San Francisco,
2003兲. We recommend Hartle’s advanced undergraduate text
to beginners and experts alike. It provides a thorough introduction to GR with a delightful emphasis on both physical
concepts and experimental tests. Hartle treats black holes in
some depth, discussing orbits, energy extraction, and the detailed solutions. 共I兲
IV. ADVANCED MATERIAL: GENERAL
Good introductions to black-hole physics can be found in
many general-relativity textbooks, of which a large number
are available at the U.S. graduate-student level. Some of our
favorites are described below. Unless otherwise noted, these
books require a thorough understanding of undergraduatelevel classical mechanics but do not presume knowledge of
more advanced material. Sections V–VIII provide a guide to
particular topics in black-hole physics and include a few
more specialized texts.
42. A First Course in General Relativity, B. F. Schutz
共Cambridge U. P., Cambridge, 1985兲. A text at the advanced
undergraduate or early graduate-level with an excellent treatment of stress-energy tensors and a brief introduction to
black-hole physics. 共I/A兲
43. Spacetime and Geometry: An Introduction to General Relativity, S. Carroll 共Benjamin Cummings, San Francisco, 2003兲. A modern beginning graduate-level text that
emphasizes links to field theory. Carroll’s two chapters on
black-hole physics provide a readable introduction to the
subject. 共A兲
44. General Relativity, R. M. Wald 共U. of Chicago Press,
Chicago, 1984兲. Wald’s book is the standard text for a
mathematical-physics-based advanced graduate course on
general relativity. 共A兲
45. Gravitation, C. Misner, K. S. Thorne, J. A. Wheeler
共W.H. Freeman and Company, New York, 1973兲. Misner,
Thorne, and Wheeler provide an encyclopedic reference for
GR as of the early 1970s. 共A兲
46. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, S. Weinberg
共Wiley, New York, 1972兲. A good text for those who wish to
do calculations. Weinberg’s book emphasizes equations over
geometric intuition and provides a solid connection to cosmology. 共A兲
47. The Large Scale Structure of Space-Time, S. W.
Hawking and G. F. R. Ellis 共Cambridge U. P., Cambridge
1973兲. The classic reference on causal structures, singularity
theorems, and mathematical physics aspects of general relativity. 共A兲
V. ASTROPHYSICAL BLACK HOLES
It is now thought that almost all galaxies contain “supermassive” black holes at their centers, millions or even billions of times more massive than the Sun. Some of these
objects power the most energetic 共nonexplosive兲 phenomena
in the universe, such as AGN and quasars. Others, like the
black hole at the center of the Milky Way, are much more
298
Am. J. Phys., Vol. 77, No. 4, April 2009
quiet. Galaxies are also thought to contain many examples of
stellar-mass black holes, with masses a few times greater
than that of the Sun, which are the evolutionary end point of
very massive stars.
While we cannot observe black holes directly, we can observe and measure their effects on the surroundings. In fact,
black holes are extremely efficient at converting the gravitational potential energy of in-falling gas into radiation. Except
under special circumstances, the gas that falls into a black
hole 共be it interstellar/intergalactic gas, or supplied by a companion star兲 does not plunge in directly; instead it forms
what is known as an accretion disk. The inner disks of supermassive black holes reach thousands of degrees Kelvin,
while stellar-mass black holes can heat their disks to millions
of degrees, where they emit in the x-ray part of the spectrum.
After listing a few basic texts in Sec. V A, in Secs.
V B–V D we organize other resources by whether the relevant black holes are stellar mass, supermassive, or of intermediate mass scale 共between the former two兲. Material concerned with specific astrophysical sources, most notably the
supermassive black hole in the center of our own galaxy,
appears separately in section V E.
A. Astrophysics: Basic textbooks and reviews
See Ref. 4 from Sec. II A.
48. “Probes and Tests of Strong-Field Gravity with Observations in the Electromagnetic Spectrum,” D. Psaltis, Living
Rev. Relativ., 11共9兲, e-print arXiv:0806.1531. This review
serves as a nexus between the previous sections of this Resource Letter and the following ones, which cover the observable aspects of black holes. Psaltis summarizes the current prospects for using astrophysical sources hosting black
holes as tools for testing the predictions of GR in the strongfield regime, and does so in a concise and clear manner. 共I兲
49. Accretion Power in Astrophysics, J. Frank, A. King,
and D. Raine 共Cambridge U. P., Cambridge, 2002兲. Assuming a basic knowledge of physics, the authors examine the
process of accretion, i.e., the extraction of gravitational potential energy from matter falling onto a compact object. A
fundamental textbook for astronomy graduate students. 共I/A兲
50. The Galactic Supermassive Black Hole, F. Melia
共Princeton U. P., Princeton, 2007兲. This book is an excellent
starting point for astrophysics graduate students interested in
the black hole associated with galactic center source Sagittarius A*. The history of observations is described along
with the observational signatures themselves. These observations range from radio to x-ray wavelengths. Nearby stellar
orbits, accretion, flares, and lensing are discussed. While primarily written from the astrophysical point of view, the text
devotes one chapter to explaining elements of GR that are
key to understanding black holes in general. 共A兲
51. “Fluorescent iron lines as a probe of astrophysical
black hole systems,” C. Reynolds and M. Nowak, Phys.
Rep., 377, 389–466 共2003兲; e-print arXiv:astro-ph/0212065.
As gas accretes onto a black hole, its velocity approaches the
speed of light. This should give rise to observable relativistic
effects, such as gravitationally redshifted emission lines. In
this extensive review, Reynolds and Nowak describe relativistic iron lines as probe of the innermost regions around accreting black holes. It also features a broad, pedagogical introduction. 共I兲
52. “Black holes and relativistic jets,” R. D. Blandford,
Prog. Theor. Phys. Suppl. 143, 182–201 共2001兲; e-print
E. Gallo and D. Marolf
298
arXiv:astro-ph/0110394. Accreting black holes of all mass
scales seem to be able to form narrow, bipolar streams of
outflowing matter/energy, as known as jets. Blandford discusses a variety of viable mechanisms through which mass,
angular momentum, and energy can escape the pull of gravity and be ejected outwards at relativistic velocities. Particular attention is paid to the connection between the jet, the
accretion disk, and the black hole itself. Written by a world
leader, as well as perhaps the most cited author共ity兲, in the
field. 共I/A兲
B. Stellar-mass black holes
53. Black Holes, White Dwarfs and Neutron Stars: The
Physics of Compact Objects, S. L. Shapiro and A. Teukolsky 共Wiley, New York, 1983兲. This excellent textbook brings
together different branches of physics—from nuclear physics, to hydrodynamics, to special relativity and GR—to understand the birth, evolution, and death of a massive star
under the special circumstances that lead to the formation of
a binary star system hosting a relativistic compact object. 共A兲
54. “Black holes in binary systems. Observational appearance,” N. I. Shakura and R. A. Sunyaev, Astron. Astrophys.
24, 337–355 共1973兲. In this seminal paper, Shakura and Sunyaev explain how, in a binary system made of an evolved
star that transfers part of its envelope mass to a black hole,
the outward transfer of angular momentum leads to the formation of a disk around the black hole. 共A兲
55. “Image of a spherical black hole with thin accretion
disk,” J.-P. Luminet, Astron. Astrophys. 75, 228–235 共1979兲.
The first investigation of the actual optical appearance of a
spherical black hole surrounded by thin accretion disk; the
spectral shifts arising from gravitational and Doppler shifts
result in strong asymmetry in the flux distribution. This calculation can be generalized to black holes of any mass. 共A兲
56. “Black hole binaries,” J. E. McClintock and R. A.
Remillard, in “Compact Stellar X-Ray Sources,” edited by
W. Lewin and M. van der Klis 共Cambridge U. P., Cambridge
2006兲, pp. 157–214; e-print arXiv:astro-ph/0306213. The
reader is introduced to the 40+ known systems hosting candidate stellar-mass black holes in the Milky Way. About half
of them are known to contain an object whose mass exceeds
the maximum theoretical mass for a neutron star, and are
referred to as “dynamically confirmed” black hole accretors.
共I兲
57. “Relativistic jets from x-ray binaries,” R. P. Fender, in
“Compact Stellar X-Ray Sources,” edited by W. Lewin and
M. van der Klis 共Cambridge U. P., Cambridge 2006兲, pp.
381–420; e-print arXiv:astro-ph/0303339. Relativistic jets
are among the most spectacular objects in the universe, and
are increasingly being found throughout the Milky Way. Despite being discovered in the radio band, they have been
proven to emit across the entire electromagnetic spectrum.
This review deals with the variety of jets from galactic
stellar-mass black holes 共as well as neutron stars兲, and their
connection with the accretion flow. 共I兲
58. “Advection-dominated accretion and black hole event
horizons,” R. Narayan and J. E. McClintock, New Astron.
Rev. 51共10–12兲, 733–751; e-print arXiv:astro-ph/0803.0322.
This work summarizes the observational evidence for the
presence of an event horizon in stellar-mass black holes by
comparing the luminosities of black holes and neutron stars
accreting mass at low rates. If the accretion flow has a low
enough density, the gravitational potential energy can remain
299
Am. J. Phys., Vol. 77, No. 4, April 2009
trapped in the flow. In the case of a neutron star, half of that
energy will be released as x-rays upon impact on the star
surface. Instead, the energy will be advected across the event
horizon in the case of a black hole. Indeed, this luminosity
gap has been observed in x-rays. 共A兲
59. “Gamma-ray bursts from stellar mass accretion disks
around black holes,” S. Woosley, Astrophys. J. 405, 273–277
共1993兲. Gamma-ray bursts are the most energetic, explosive
astrophysical events we know of. In a flash lasting few ticks
of a clock, more energy is released than over the entire Sun
lifetime. Woosley’s model for the production of 共long兲
gamma-ray bursts involves the formation of a stellar-mass
black hole as the evolutionary endpoint of Wolf-Rayets, i.e.,
a particular class of massive stars. 共A兲
C. Supermassive black holes
60. Active Galactic Nuclei: From the Central Black
Hole to the Galactic Environment, J. Krolik 共Princeton U.
P., Princeton, 1998兲. A comprehensive description of the observational properties of supermassive black holes that are
actively accreting at the center of AGN. A must-read resource for astronomers, physicists interested in applications
of the theory of gravitation, and graduate students. 共E/I兲
61. “Black hole models for active galactic nuclei,” M. J.
Rees, Annu. Rev. Astron. Astrophys. 22, 471–506 共1984兲. An
inspired—and inspiring—work that reviews the body of evidence for supermassive black holes as the power engine for
AGN: quasars, radio galaxies, blazars, and related objects. 共I兲
62. “Variability of active galactic nuclei,” B. M. Peterson,
in Advanced Lectures on the Starburst-AGN Connection,
edited by I. Aretxaga, D. Kunth and R. Mújica 共World Scientific, Singapore, 2001兲 pp. 3–68. A comprehensive description of “reverberation mapping” techniques, which allow one
to infer the masses of black holes in active galaxies at high
redshifts. This method is based on measuring the time delay
between variations in the line and continuum emission from
distant AGN. The delay provides an indirect measure of the
distance between the supermassive black holes and the regions where the emission lines are being excited. This characteristic scale size constrains the size of the black hole, and
thus its mass. 共I兲
63. “Quasars and galaxy formation,” J. Silk and M. J.
Rees, Astron. Astrophys. 331, L1–L4 共1998兲; e-print
arXiv:astro-ph/9801013. A visionary paper on the interplay
between the formation of black-hole seeds in the early universe and the evolution of 共proto-兲galaxies. 共I/A兲
64. “Supermassive black holes in galactic nuclei: Past,
present and future research,” L. Ferrarese and F. Holland,
Space Sci. Rev. 116, 523–624, 共2005兲; e-print arXiv:astroph/0411247. This review discusses the current status of supermassive black-hole research from an observational perspective. Recent observations have unveiled empirical
scaling relations between the mass of the central black hole
and the global properties of the host galaxy 共see Refs. 78 and
79兲, suggesting that black holes influence the very formation
and evolution of cosmic structures since the dawn of time. 共I兲
65. “Massive black holes: Formation and evolution” M. J.
Rees and M. Volonteri, in Black Holes From Stars to
Galaxies—Across the Range of Masses, edited by V. Karas
and G. Matt 共Cambridge U. P., Cambridge, 2007兲; e-print
arXiv:astro-ph/0701512. Rees and Volonteri summarize the
progresses and open questions about the formation and
growth of supermassive black holes, as well as the prospects
E. Gallo and D. Marolf
299
for using them as laboratories for testing the strong gravity
regime. This work includes a list of the most important 共albeit advanced兲 bibliographic references in the field. 共I/A兲
66. Energy input from quasars regulates the growth and
activity of black holes and their host galaxies, T. Di Matteo,
V. Springel, L. Hernquist, Nature 共London兲 433, 604–607
共2005兲; e-print arXiv:astro-ph/0502199. The evolution of
galaxies along with their nuclear supermassive black holes
has been followed by means of detailed numerical simulations. It is found that a galaxy-galaxy merger leads to a massive enhancement of the inner accretion rate, enough to
switch on the nuclear black hole as a quasar. In turn, the
energy release by the quasar is able to quench star formation
and halt further growth. This self-regulatory mechanism sets
the quasars’ duty cycle and may be able to explain the empirical scaling relations Refs. 78 and 79 共I兲
67. “The many lives of active galactic nuclei: Cooling
flows, black holes and the luminosities and colors of galaxies,” D. J. Croton et al., Mon. Not. R. Astron. Soc., 365,
11–28 共2006兲; e-print arXiv:astro-ph/0508046. Energy “feedback” from supermassive black holes could solve a number
of problems faced by the hierarchical paradigm for the formation and evolution of cosmic structures at galactic and
subgalactic scales, such as the observed red colors of massive spheroidal galaxies. Croton et al. couple the output from
state-of-the-art dark-matter simulations to semianalytical
modeling to show that nuclear supermassive black holes may
be able to quench the formation of young blue stars by injecting heat and mechanical power into the surroundings. In
order for black-hole feedback to be effective, low levels of
prolonged activity are needed, as opposed to sudden bursts
of power. These effects should be observable in the cores of
nearby, formally inactive galaxies. 共A兲
68. “Massive black hole binary evolution,” D. Merritt and
M. Milosavljević, Living Rev. Relativ. 共2005兲, http://
relativity.livingreviews.org/Articles/lrr-2005-8/. This article
reviews the observational evidence 共or lack thereof兲 for binary supermassive black holes and discusses their evolution
and final fate. Although a compelling case for a bound binary
system is yet to be discovered, a number of sources are believed to host two supermassive black-hole binaries at projected separations of less than 3000 light years. 共A兲
69. “LISA observations of rapidly spinning massive black
hole binary systems,” A. Vecchio, Phys. Rev. D 70 042001,
共2004兲; e-print arXiv:astro-ph/0304051. The Laser Interferometer Space Antenna 共LISA: lisa.nasa.gov/兲 is a 共planned兲
space-based gravitational-wave observator. Detecting gravitational waves from supermassive blackhole binaries will
have tremendous impact on our understanding of gravity in
the highly nonlinear relativistic regime, as well as the formation and evolution of galaxies in a cosmological context.
This review focuses on the effects of black-hole spin on the
expected LISA gravitational-wave signal, and describes how
accounting for these effects can substantially reduce the uncertainties on other parameters, such as the reduced mass.
共A兲
D. Intermediate mass black holes
70. “Ultraluminous x-ray sources in external galaxies,” A.
R. King et al., Astrophys. J. 552, L109–L112 共2001兲; e-print
arXiv:astro-ph/0104333. With the launch of sensitive x-ray
observatories, a new population of compact, luminous, offnuclear x-ray sources has been unveiled in external galaxies.
300
Am. J. Phys., Vol. 77, No. 4, April 2009
While these so called “ultraluminous x-ray sources” are natural intermediate-mass black-hole candidates, several—less
exotic—explanations are offered in this paper. 共I/A兲
71. “A comparison of intermediate-mass black hole candidate ultraluminous x-ray sources and stellar-mass black
holes,” J. M. Miller, A. C. Fabian, and M. C. Miller, Astrophys. J., 614, L117–L120 共2004兲; e-print arXiv:astro-ph/
0406656. On the same topic as the previous work, this paper
focuses instead on the observational evidence-for intermediate mass black holes in a small number of ultraluminous
x-ray sources, namely those where the x-ray continuum spectra appear too cold to be due to accreting stellar-mass black
holes. 共I/A兲
72. “X-ray observations of ultraluminous x-ray sources,”
T. P. Roberts, Astrophys. Space Sci. 311, 203 共2007兲; e-print
arXiv:astro-ph/0706.2562. A brief, critical review on the
X-ray properties of ultraluminous x-ray sources, and their
possible physical nature. 共I/A兲
E. Black holes: case studies
Throughout this section, we list a number of journal papers that report on observational milestones on specific
black-hole sources, or an ensemble of them. The two last
entries are devoted to the case for a 3–4 million solar-mass
black hole at the center of our own Milky Way. This list is
not meant to be complete, but simply to offer a glance at
groundbreaking astrophysical discoveries.
73. “Observation of a correlated x-ray transition in Cygnus
X-1,” H. Tananbaum et al., Astrophys. J. 177, L5–L5 共1972兲.
X-ray astronomy began only in the early 1960s. The first
rocket flight to detect a cosmic x-ray source was launched in
1962 by a group at American Science and Engineering. In
the 1970s, the dedicated x-ray satellite “Uhuru” repeatedly
observed a newly discovered x-ray source in the Cygnus
constellation over a period of 16 months. Its x-ray flux was
found to vary significantly, and in correlation with its radio
counterpart. The first “black-hole state transition” 共see Ref.
56兲 had just been discovered. 共I兲
74. “X-ray fluorescence from the inner disc in Cygnus
X-1,” A. C. Fabian et al., Mon. Not. R. Astron. Soc. 238,
729–736 共1989兲. It is argued that the broad emission line in
the x-ray spectrum of the 10-solar-mass black-hole candidate
in Cygnus X-1 is well accounted for by a model of fluorescent emission from the inner parts of an inclined accretion
disk. 共I/A兲
75. “A superluminal source in the galaxy,” I. F. Mirabel
and L. F. Rodriguez, Nature 共London兲 371, 46–48 共1994兲. A
seminal paper reporting on the discovery of radio-emitting
plasmons traveling with superluminal velocities 共apparent
velocities greater than the speed of light兲 within the Milky
Way. The accepted explanation is that the plasmons are
ejected in opposite directions from the central stellar-mass
black hole at speeds higher than 0.7c. If the ejection direction is close to the observer’s line of sight, this can lead to
apparent superluminal velocities. 共E/I兲
76. “Gravitationally redshifted emission implying an accretion disk and massive black-hole in the active galaxy
MCG 6-30-15,” Y. Tanaka et al., Nature 共London兲 375, 659–
661 共1995兲. Tanaka and collaborators report on the discovery
of relativistic effects in an x-ray emission line from ionized
iron in the galaxy MCG 6-30-15. The line is extremely broad
and asymmetric, with most of the line flux being redshifted.
共I/A兲
E. Gallo and D. Marolf
300
77. “Formation of the radio jet in M87 at 100 Schwarzschild radii from the central black hole,” W. Junor, J. A. Biretta, and M. Livio, Nature 共London兲 401, 891–892 共1999兲.
This work reports on high-resolution radio observations of
the inner regions of the nearby active galaxy M87, which
powers a highly collimated jet. The jet radio maps are consistent with the hypothesis that jets are formed by an accretion disk around the central black hole, which is threaded by
a magnetic field. 共I兲
78. “Inward bound—The search for supermassive black
holes in galactic nuclei,” J. Kormendy and D. Richstone,
Annu. Rev. Astron. Astrophys 33, 581–581 共1995兲. The discovery of an empirical scaling relation between the mass of
nuclear supermassive black holes and the mass of the host
galaxy’s bulge. 共A兲
79. “A fundamental relation between supermassive black
holes and their host galaxies,” L. Ferrarese and D. Merritt,
Astrophys J. 539, L9–L12 共2000兲; e-print arXiv:astro-ph/
0006053, “A relationship between nuclear black hole mass
and galaxy velocity dispersion,” K. Gebhardt et al., Astrophys. J. 539, L13–L16 共2000兲; e-print arXiv:astro-ph/
0006289. The discovery 共by two different groups兲 of an empirical scaling relation between the mass of nuclear
supermassive black holes and the velocity dispersion of stars
orbiting much farther than the black-hole sphere of influence.
共A兲
80. “A 20,000 solar mass black hole in the stellar cluster
G1,” K. Gebhardt, R. M. Rich, and L. C. Ho, Astrophys. J.
578, L41–L46 共2002兲; e-print arXiv:astro-ph/0209313. At
the center of a globular cluster, this object represents the
strongest intermediate-mass black-hole candidate to date.
Unlike in the examples listed in Sec. V D, its presence is
inferred from dynamics rather than from electromagnetic signals. 共A兲
81. “The nucleus of our galaxy,” R. Genzel, D. Hollenbach, and C. H. Townes, Rep. Progr. Phys. 57, 417–479
共1994兲. The subject of this review is the central 100 parsecs
of our galaxy, with a strong focus on the central few parsecs.
The authors discuss the stellar and interstellar components,
the importance of magnetic and gravitational forces, the evidence for stellar formation and a central massive black hole,
and the origin and nature of ionization, outflows, and
interstellar-gas dynamics. 共I/A兲
82. “High proper-motion stars in the vicinity of Sagittarius
A*: Evidence for a supermassive black hole at the center of
our galaxy,” A. Ghez, B. L. Klein, M. Morris, and E. E.
Becklin, Astrophys. J. 509, 678–686 共1998兲; e-print
arXiv:astro-ph/9807210. The description of a groundbreaking astrophysics experiment: The motion of 90 faint stars in
the vicinity of the galactic center has been followed over a
period of two years with the Keck 10 m telescope. Their
motion allows for the determination of the enclosed mass
through Kepler’s laws. A central dark mass of about 3 million solar masses is confined to an area about 10 times
smaller than Earth’s orbit around the Sun. The inferred density leads us to the conclusion that our galaxy harbors a
massive central black hole. 共A兲
83. “Evidence for a black hole from high rotation velocities in a sub-parsec region of NGC4258,” M. Miyoshi, J.
Moran, J. Herrnstein, L. Greenhill, N. Nakai, P. Diamond,
and M. Inoue, Nature 共London兲 373, 127–129 共1995兲. Along
with the nucleus of the Milky Way 共Refs. 81 and 82兲, this
galaxy presents the best known case for a super massive
black hole. Here the central dark mass is weighed by recon301
Am. J. Phys., Vol. 77, No. 4, April 2009
structing the line-of-sight velocity profile of water-maseremitting gas orbiting around it. Since water-maser emission
is a narrow feature, the line-of-sight velocity can be determined from the Doppler shift variations of the observed maser frequency. The emission regions trace Keplerian orbits
which allow one to estimate the enclosed mass as a function
of radius, and hence the dark mass density. NGC4258 is
thought to host a 40 million solar mass black hole. 共I兲
F. Image galleries: Black holes from space
This is all well and fine, one may say. But what is it that
we actually see when we go up there, outside of Earth’s
atmosphere? Browsing the following websites, the reader
will be taken through a journey of the universe as seen by
NASA’s three “Great Observatories.”
84.
hubblesite.org/gallery/album/exoticគcollection/
blackគhole/
Space Telescope Science Institute/HST Gallery: Black Holes.
An amazing gallery of AGN pictures, taken by one of the
most powerful optical eyes ever built. NASA’s crown jewel,
the Hubble Space Telescope, has provided us with a window
to the unexplored universe since 1990. 共E兲
85. chandra.harvard.edu/photo/category/blackholes.html
Chandra X-Ray Observatory Photo Album: Black Holes.
Black holes shine in the x-ray band because of the high temperatures attained by the in-falling gas; however, x-rays from
space are absorbed by the Earth atmosphere. Since its launch
in 1999, the Chandra X-Ray Observatory has contributed
substantially to our understanding of accreting black holes in
the Milky Way as well as external galaxies. This friendly site
is a collage of Chandra images, all equipped with accessible
explanations. 共E兲
86. gallery.spitzer.caltech.edu/Imagegallery/subcat.php?cat
⫽AstronomicalគImages&subcat⫽GalaxiesគandគtheគUniverse
Spitzer Space Telescope Images: Galaxies and the Universe.
The last of NASA’s Great Observatories to fly, Spitzer is a
flying infrared telescope: UV/x-rays from accreting supermassive black holes can be partly absorbed by surrounding
dust and re-emitted at lower frequencies. Here is how AGN
would look if human eyes were sensitive to the same frequencies as microwaves. 共E兲
VI. STATICS, DYNAMICS, AND
THERMODYNAMICS OF BLACK HOLES
This section addresses the well-established mathematical
physics of black holes. We have attempted to impose some
structure on this highly interconnected body of knowledge
by dividing the material as follows: Section VI A deals with
the statics of black holes, including the construction of stationary solutions, no-hair and uniqueness theorems, some
work on hairy black holes, and definitions of horizons. Perturbations, quasinormal modes, and stability of stationary solutions is addressed in Sec. VI B. We have chosen to include
the important subject of extreme-mass in-spirals and the calculation of gravity-wave production by post-Newtonian expansions in this section. Section VI C then considers the
strongly nonlinear dynamics of black holes associated with
critical phenomena or with black-hole collisions. Most of our
discussion of numerical GR can be found here, as is the
associated material on gravity waves. Section VI D contains
resources on black-hole thermodynamics, quantum-field
theory in curved spacetime, and the associated description of
E. Gallo and D. Marolf
301
Hawking radiation. More microscopic treatments of Hawking radiation and black-hole entropy are postponed to Sec.
VII. Section VI A–VI D focus primarily on black holes in
d = 4 spacetime dimensions; resources on black holes in other
spacetime dimensions have been collected in Sec. VI E. We
have chosen to include discussions of possible black-hole
production at particle colliders in this latter section.
The reader should be aware of a few peculiarities that
result from this organization. Resources on gravity-wave
production by black holes appear in both Secs. VI B and
VI C. Resources on isolated and dynamical horizons appear
in Secs. VI A, VI C, and VI D, and also in Sec. VII B. There
is also much overlap between Sec. VI E 共black holes in d
⫽ 4 dimensions兲 and Secs. VI D, VII A, and VIII, and between Sec. VI D 共thermodynamics兲 and VII C 共where we
have chosen to locate discussions of entropy bounds and the
proposed holographic principle, as well as Jacobson’s work
共Ref. 149兲 on the Einstein equation of state兲. We apologize to
the reader for any confusion this may cause.
A. Statics
87. “No-hair Theorems: Folklore, conjectures, results,
chrusciel, P. T.,” in Differential Geometry and Mathematical
Physics, edited by J. Beem, and K. L. Duggal., Am. Math.
Soc., Providence, Vol 170, pp. 23–49 共1994兲; e-print
arXiv:gr-qc/9402032. “Uniqueness of Stationary, ElectroVacuum Black Holes Revisited,” P. T. Chrusciel, Helv. Phys.
Acta 69, 529–552 共1996兲. For a related online version see:
“Uniqueness of Stationary, Electro-Vacuum Black Holes Revisited,” P. T. Chrusciel 共October, 1996兲, e-print arXiv:gr-qc/
9610010. An important property of black holes is that, when
left to themselves, they settle down to solutions described by
only a few parameters, most importantly their mass M and
angular momentum J. In other words, the surface of a black
hole must be very smooth; even in principle, the surface of a
black hole cannot support mountains, valleys, or other complicated features. One says that black holes “have no hair.”
These papers review and complete classic work on the nohair theorems. 共A兲
88. “Stationary black holes: Uniqueness and beyond,”
Markus Heusler, Living Reviews in Relativity 共1998兲,
relativity.livingreviews.org/Articles/lrr-1998-6/index.html. In
sufficiently complicated settings, the number of parameters
needed to describe black holes increases and black holes can
support a limited amount of “hair.” This review describes the
original no-hair theorems as well as some of the more complicated hairy black holes. See also Ref. 97 for more on
nonabelian black holes. 共A兲
89. “Black hole boundaries,” I. Booth, Can. J. Phys. 83,
1073–1099 共2005兲; e-print arXiv:gr-qc/0508107. Black holes
are traditionally defined as regions of space-time from which
in principle signals cannot propagate to infinity. Such definitions clearly break down in cosmological settings, where appropriate asymptotic regions may not exist. Booth reviews
recent work on alternative “quasilocal” definitions of blackhole horizons. See also Refs. refIH1, 106, and 136. 共A兲
B. Perturbative black-hole dynamics
90. “Quasi-normal modes of stars and black holes,” K. D.
Kokkotas and B. Schmidt, Living Reviews in Relativity
共1999兲, http://relativity.livingreviews.org/Articles/lrr-19992/. When a black hole is perturbed, it “rings” with a charac302
Am. J. Phys., Vol. 77, No. 4, April 2009
teristic set of frequencies. However, the amplitude at each
frequency decays as energy falls through the horizon and
radiates to infinity. As a result, the solutions associated with
this ringing are called “quasinormal” modes. Kokkotas and
Schmidt review such perturbations for both black holes and
relativistic stars, and in the process review the stability of
black holes. See also Ref. 91. 共A兲
91. “Quasi-normal modes and gravitational wave astronomy,” V. Ferrari and L. Gualtieri, Gen. Relativ. Gravit.
40, 945–970 共2008兲; e-print arXiv:0709.0657 关gr-qc兴. A review of quasi-normal modes with emphasis on possible detection of the associated gravitational waves. See also Ref.
90 共A兲
92. “Extreme Mass ratio inspirals: LISA’s unique probe of
black hole gravity,” K. Glampedakis, Class. Quantum Grav.
22, S605–S659 共2005兲; e-print arXiv:gr-qc/0509024. The
proposed LISA gravitational-radiation detector is expected to
be sensitive to gravity waves produced when stars or stellarmass black holes spiral into supermassive black holes. In
such cases the extreme ratio between the two black-hole
masses provides a small parameter that allows controlled calculations of both these gravitational waves and the backreaction on the small body’s orbit. This review begins with
free-particle motion on the black-hole background and builds
up toward more complicated calculations. 共A兲
93. “Analytic black hole perturbation approach to gravitational radiation,” M. Sasaki and H. Tagoshi, Living Rev.
Relativ. 共2003兲, relativity.livingreviews.org/Articles/lrr2003-6/index.html. A slightly more advanced review of
extreme-mass in-spirals describing both the relevant perturbative and post-Newtonian techniques. 共A兲
94. “Resource Letter GrW-1: Gravitational waves,” J. M.
Centrella, Am. J. Phys. 71, 520–525 共2003兲; e-print arXiv:grqc/0211084. Centrella’s Resource Letter is an excellent guide
to the pre-2003 literature on the production of gravity waves
by events involving black holes. 共E/I/A兲
95. Black Holes: The Membrane Paradigm, by K. S.
Thorne, D. A. MacDonald, and R. H. Price 共Yale U. P. , New
Haven 1986兲. This compilation of classic papers shows that
black-hole horizons respond to outside perturbations like a
two-dimensional viscous fluid or membrane. The horizon
conducts electricity but 共at least classically兲 does not conduct
heat. As a result, horizons are described by both the Navier–
Stokes equations and Ohm’s law. 共A兲
96. “The internal structure of black holes, “W. Israel, in
Black Holes and Relativistic Stars, edited by R. Wald 共U. of
Chicago Press, Chicago, 1998兲, pp. 137–151. Analytic solutions for charged and rotating black holes contain a so-called
Cauchy horizon at which predictability appears to break
down. However, this Cauchy horizon is unstable to the formation of a singularity 共where quantum effects clearly become relevant兲. This concise summary of black-hole interiors
provides a good introduction and guide to the pre-1998 literature on Cauchy-horizon instabilities and the resulting phenomenon of mass inflation. See also “The interior of charged
black holes and the problem of uniqueness in general relativity,” M. Dafermos, Commun. on Pure App. Math., 58
445–504, 共2005兲; e-print gr-qc/0307013. 共A兲
97. Internal Structure of Black Holes and Spacetime Singularities: Proceedings, Annals of the Israel Physical Society
Vol. 13, edited by L. M. Burko and A. Ori 共Institute of Physics Publishing, London, 1998兲. This volume contains a number of excellent articles on subjects ranging from CauchyE. Gallo and D. Marolf
302
horizon instabilities 共in diverse contexts兲 to the internal
structure of non-abelian black holes. 共A兲
C. Nonlinear black-hole dynamics: Numerical GR and
Black-Hole Collisions
98. “Critical Phenomena in gravitational collapse,” C.
Gundlach, Living Rev. Relativ. 共1999兲, relativity.
livingreviews.org/Articles/lrr-1999-4/index.html. In 1993,
Choptuik discovered a kind of critical phenomenon associated with black-hole formation in certain contexts. For
spherically symmetric scalar fields, he considered generic
1-parameter families of initial data that interpolate between
weakly gravitating data 共which does not form a black hole兲
and strongly gravitating data 共which does兲. At the threshold
of black-hole formation one finds a complicated, singular,
scale-periodic solution while black holes of arbitrarily small
mass arise close to this threshold. Related phenomena were
later found in many systems. Gundlach reviews the modern
understanding of this field. 共A兲
99. “Binary black hole coalescence,” F. Pretorius, e-print
arXiv:0710.1338 关gr-qc兴. The coalescence of two black holes
of comparable mass is a difficult problem, best treated by
numerical techniques. 共See Refs. 92 and 93 for the case of
extreme-mass ratios.兲 After many years of effort, such methods have recently become quite successful at calculating the
properties of the final black hole and producing detailed
waveforms for gravitational radiation. Pretorius reviews
these techniques, their results, and the implications for astrophysics and other fields. 共A兲
100. “Initial Data for numerical relativity,” Greg Cook,
Living Rev. Relativ., lrr-2000-5, relativity.livingreviews.org/
Articles/lrr-2000-5/index.html Extracting useful information
from numerical simulations requires starting with appropriate initial data. Cook describes the current methods and their
limitations, with an eye toward future progress. 共A兲
101. “Introduction to isolated horizons in numerical relativity,” O. Dreyer, B. Krishnan, D. Shoemaker, and E.
Schnetter, Phys. Rev. D 67, 024018, 共2003兲; e-print arXiv:grqc/0206008. Dreyer et al. describe methods for using
isolated-horizon techniques 共see Refs. 89, 105, 106, and 136兲
to compute the mass M and angular momentum J of black
holes in numerical simulations. 共A兲
D. Black-hole thermodynamics and Hawking radiation
See also Refs. 117, 125–127
102. “The four laws of black hole mechanics,” J. M.
Bardeen, B. Carter, and S. W. Hawking, Commun. Math.
Phys. 31, 161–170, 共1973兲. This seminal paper derives a
“first law of black-hole mechanics” with striking similarities
to the first law of thermodynamics. The authors argue that
black holes also satisfy analogues of the 0th, 2nd, and 3rd
laws of thermodynamics. The idea that black holes are thermodynamic systems is now widely regarded as a fundamental clue to the nature of quantum gravity. 共A兲
103. “Black holes and entropy,” J. D. Bekenstein, Phys.
Rev. D, 7, 2333–2346 共1973兲. “Generalized second law of
Thermodynamics in black-hole Physics,” J. D. Bekenstein,
Phys. Rev. D 9, 3292–3300 共1974兲. These foundational
works suggest that the area of a black hole measures its
entropy and formulate the generalized second law of thermodynamics: that the summed entropy of matter plus black
holes should not decrease. 共A兲
303
Am. J. Phys., Vol. 77, No. 4, April 2009
104. “The thermodynamics of black holes,” R. Wald, Living Rev. Relativ. 4, 46 共2001兲; e-print arXiv:gr-qc/9912119;
relativity.livingreviews.org/Articles/lrr-2001-6/index.html.
Wald provides a comprehensive survey of black-hole thermodynamics and the generalized second law. A detailed
guide to the literature is presented, to which the reader is
referred for derivations. Hawking radiation and associated
issues are also discussed. 共I/A兲
105. “Black hole mechanics,” A. Ashtekar, in Encyclopedia of Mathematical Physics, 共Elsevier, New York, 2006兲;
300–305. Also available at cgpg.gravity.psu.edu/people/
Ashtekar/articles/bhm.pdf. Ashtekar briefly reviews the standard formulation of black-hole mechanics and then describes
the more local isolated- and dynamical-horizon formulations.
See also Refs. 89, 106, and 136. 共A兲
106. “Isolated and dynamical horizons and their applications,” A. Ashtekar and B. Krishnan, Living Rev. Relativity,
lrr-2004-10.
relativity.livingreviews.org/Articles/lrr-200410/. This more thorough review begins with the basics of the
isolated and dynamical horizon quasi-local formalism for
black-hole thermodynamics, but proceeds to address applications in numerical GR, mathematical physics, and quantum
gravity. 共A兲
107. “Particle creation by black holes,” S. W. Hawking,
Commun. Math. Phys. 43, 199–220 共1975兲. Hawking shows
that, due to quantum effects, black holes emit thermal radiation at 共very low兲 temperatures set by their so-called surface
gravity. This is the original paper on such Hawking radiation.
The calculation identifies the black-hole entropy with A / 4ᐉ2p
where A is the event-horizon area and ᐉ p is the so-called
Planck length. 共A兲
108. “Introduction to quantum fields in curved spacetime
and the Hawking effect,” T. Jacobson in Lectures on Quantum Gravity, edited by A. Gomberoff and D. Marolf
共Springer, New York, 2005兲; e-print arXiv:gr-qc/0308048.
Based on lectures given at a summer school, Jacobson’s
notes provide a very physical introduction to quantum fields
in curved spacetime and the Hawking effect. This is an excellent reference for students new to the subject. Cosmological particle creation, transplanckian questions, and other issues are also addressed. The focus is on Lorentzian
techniques. 共A兲
109. “Black hole thermodynamics,” S. F. Ross, e-print
arXiv:hep-th/0502195. Also based on lectures at a summer
school, Ross’s notes are a good complement to Ref. 108
above. This work includes a brief introduction to black holes
and a review of classical black-hole thermodynamics. Ross
also discusses quantum-field theory in curved spacetime, but
emphasizes Euclidean techniques. 共A兲
110. Quantum Fields in Curved Space, N. D. Birrell and
P. C. W. Davies 共Cambridge U. P., Cambridge, 1982兲. A
classic textbook on quantum-field theory in curved spacetime. This text begins with the basics and works up through
the derivation of Hawking radiation. 共A兲
111. Quantum Field Theory in Curved Spacetime and
Black Hole Thermodynamics, R. M. Wald 共U. of Chicago
Press, Chicago, 1994兲. This brief text quickly summarizes
these fields as of 1994. The approach is somewhat more
mathematical than Refs. 108–110 共A兲
112. “Path integral derivation of black hole radiance,” J.
B. Hartle and S. W. Hawking, Phys. Rev. D 13, 2188–2203
共1976兲; “Action integrals and partition functions in quantum
gravity,” G. Gibbons and S. W. Hawking, Phys. Rev. D, 15,
2752–2756 共1977兲. Together, these two papers introduce EuE. Gallo and D. Marolf
303
clidean techniques into black-hole thermodynamics. Hartle
and Hawking use analytic continuation to derive Hawking
radiation, while Gibbons and Hawking introduce the use of
Euclidean solutions to approximate the gravitational partition
functions. 共A兲
113. “On the origin of the particles in black hole evaporation,” R. Schutzhold and W. G. Unruh, e-print
arXiv:0804.1686 关gr-qc兴; The above references derive Hawking radiation within the framework of locally Lorentzinvariant quantum-field theory on a fixed background spacetime. This is certainly not the complete description of our
universe, and some modification of Hawking radiation is expected. The effect of breaking the local Lorentz symmetry
has been explored in some detail, and Schutzhold and Unruh
give the best understanding to date. The present evidence 共as
does Ref. 108 above兲 is that Hawking radiation is essentially
a low-energy phenomenon, and should not receive significant
corrections. 共A兲
114. “Do black holes radiate?,” A. D. Helfer, Rept. Prog.
Phys. 66, 943–1008, 共2003兲; e-print arXiv:gr-qc/0304042. As
a counterpoint to Ref. 113, we include Helfer’s review of
arguments that Hawking radiation might receive large corrections from quantum-gravity effects. The reader should
note, however, that certain arguments in this work are out of
date in view of Refs. 108 and 113. 共A兲
115. “Black hole entropy is the Noether charge,” R. M.
Wald, Phys. Rev. D 48, 3427–3431 共1993兲; e-print arXiv:grqc/9307038. “Some properties of Noether charge and a proposal for dynamical black hole entropy,” V. Iyer and R. M.
Wald, Phys. Rev. D 50, 846–864 共1994兲; e-print arXiv:gr-qc/
9403028. A comparison of Noether charge and Euclidean
methods for computing the entropy of stationary black holes,
Phys. Rev. D 52, 4430–4439 共1995兲; e-print arXiv:gr-qc/
9503052. These works generalize the first law of black-hole
thermodynamics to theories more complicated than just Einstein gravity plus matter. Higher derivative theories are included. 共A兲
E. Black holes in dimensions d Å 4
116. “The black hole in three-dimensional space-time,” M.
Banados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 69,
1849–1851 共1992兲; e-print arXiv:hep-th/9204099; “Geometry of the 共2 + 1兲 black hole,” M. Banados, M. Henneaux, C.
Teitelboim and J. Zanelli, Phys. Rev. D 48, 1506–1525
共1993兲; e-print arXiv:gr-qc/9302012. In three dimensions,
black holes arise only in the presence of a negative cosmological constant 共⌳ ⬍ 0兲. These so-called BTZ black holes
are particularly simple, as they are quotients of 2 + 1 anti-de
Sitter space. The above works introduce such solutions and
study their properties. 共I/A兲
117. “Dilaton gravity in two dimensions,” D. Grumiller,
W. Kummer; and D. V. Vassilevich, Phys. Rep. 369, 327–430
共2002兲; e-print arXiv:hep-th/0204253. Many calculations
simplify in 1 + 1 dimensions. While there is no meaningful
theory of Einstein gravity in 1 + 1 dimensions, the so-called
dilaton gravity theories capture some of the same physics
and can be related to Einstein gravity in higher dimensions.
This work reviews such models, with emphasis on blackhole solutions, Hawking radiation, and quantum effects. 共A兲
118. “Black holes in higher dimensions,” R. Emparan and
H. S. Reall, Living Rev. Relativ. 11共6兲, e-print
arXiv:0801.3471 关hep-th兴. The space of black-hole solutions
and behaviors becomes much richer in d ⬎ 4 dimensions
304
Am. J. Phys., Vol. 77, No. 4, April 2009
where one finds black strings, black branes, and black rings
in addition to the natural generalizations of 3 + 1 black holes.
New behaviors arise as well, as some black objects are unstable and appear to lead to violations of certain cosmic censorship conjectures. Emparan and Reall review these solutions, the techniques used to construct them, their
thermodynamics, and what is known about their dynamics.
See also Ref. 120. The section on open problems may be of
particular interest for the motivated student. 共A兲
119. “Black rings,” R. Emparan and H. S. Reall, Class.
Quantum Grav. 23, R169–R197 共2006兲; e-print arXiv:hep-th/
0608012. This more-focused review addresses both the classical physics and microscopic string theory of black rings.
共A兲
120. “Instabilities of black strings and branes,” T. Harmark, V. Niarchos, and N. A. Obers, Class. Quantum Grav.
24, R1–R90 共2007兲; e-print arXiv:hep-th/0701022. While
black holes in 3 + 1 dimensions are typically stable, black
objects in higher dimensions exhibit interesting instabilities
that may lead to violations of at least certain forms of the
cosmic censorship hypothesis. This work reviews the classic
Gregory-Laflamme instability of 4 + 1 black strings as well as
many generalizations. The correlated stability conjecture and
its limitations are discussed. As in Ref. 118, the section on
open problems may be of particular interest for the motivated
student. 共A兲
121. “The phase transition between caged black holes and
black strings: A review,” B. Kol, Phys. Rep. 422, 119–165
共2006兲; e-print arXiv:hep-th/0411240. In d ⬎ 4 dimensions,
which sort of black-hole solution is most stable can depend
on the size of the box or “cage” in which it is placed. Large
boxes favor black holes, while small boxes favor black
strings or branes. Kol reviews the complex structure associated with the associated phase transitions. 共A兲
122. “Black holes and solitons in string theory,” D. Youm,
Phys. Rep. 316, 1–232 共1999兲; e-print arXiv:hep-th/9710046.
An encyclopedic review of black-hole and black-brane solutions in string theory as of 1997. 共A兲
123. “BPS branes in supergravity,” K. S. Stelle, e-print
arXiv:hep-th/9803116, in 1997 Summer School in High
Energy Physics and Cosmology, edited by E. Gava et al.
共World Scientific, New Jersey, 1998兲, pp. 29–127. Stelle introduces brane solutions in supergravity, discussing supersymmetry, Kaluza–Klein reduction, and low-velocity scattering of branes. 共A兲
124. “String/M-branes for relativists,” D. Marolf, e-print
arXiv:gr-qc/9908045. A brief introduction to stringy branes,
intended to convey some basic aspects of brane physics and
perspectives on string theory to those trained in GR. 共E/I/A兲
125. “Black holes at future colliders and beyond: A review,” G. L. Landsberg, J. Phys. G 32, R337–R365 共2006兲;
e-print arXiv:hep-ph/0211043. If our universe has more than
four dimensions, then it may be possible to produce black
holes at particle colliders, or they may be produced naturally
in our atmosphere owing to the impacts of cosmic rays.
Landsberg reviews the status of these ideas as of 2002, including the signatures of such events that could be used to
detect them. 共A兲
126. “High-energy black hole production,” S. B. Giddings,
AIP Conf. Proc. 957, 69–78 共2007兲; e-print arXiv:0709.1107
关hep-ph兴兴. A brief but more recent review of the possible
production and detection of TeV-scale black holes. Includes a
discussion of open problems. 共A兲
127. “Black holes in theories with large extra dimensions:
E. Gallo and D. Marolf
304
A review,” P. Kanti, Int. J. Mod. Phys. A 19, 4899–4951
共2004兲; e-print arXiv:hep-ph/0402168. A more thorough review of mini black holes in models with large extra dimensions, focusing on their decay processes. 共A兲
128. “Braneworld black holes in cosmology and astrophysics,” A. S. Majumdar and N. Mukherjee, Int. J. Mod.
Phys. D 14, 1095–1129 共2005兲; e-print arXiv:astro-ph/
0503473. A review of larger black holes in models with large
extra dimensions and their astrophysical implications. 共A兲
VII. BLACK-HOLE MICROPHYSICS
The resources below address the deep microphysics of
black holes. The reader should be aware that most of these
works make certain assumptions about the quantum nature of
gravity and spacetime that are beyond the reach of current
experiments. We have attempted to sort this material on the
basis of these assumptions into string theory 共Sec. VII A兲,
loop quantum gravity 共Sec. VII B兲, and other 共Sec. VII C兲.
Section VII C includes not only other approaches to quantum
gravity, but also discussions of entropy bounds, the proposed
holographic principle, black-hole complementarity, and Jacobson’s work on the Einstein equation of state Ref. 149.
Certain overlaps and ambiguities are of course unavoidable. The string-theory material is closely related to that of
Secs. VI E 共higher dimensions兲 and VIII and has at least
historical connections to some material in Sec. VII C, while
the loop gravity work below is intimately tied to the treatments of isolated and dynamical horizons Refs. 89, 105, and
106 共which we have chosen to list again as Ref. 139兲. There
is also much overlap between Secs. VII C and VI D.
A. Black-hole microphysics in string theory
129. String Theory, Vols. 1 & 2, J. Polchinski 共Cambridge U. P., Cambridge, 1998兲. D-branes, C. Johnson
共Cambridge U. P., Cambridge, 2003兲; String Theory and
M-Theory: A Modern Introduction, K. Becker, M. Becker,
and J. Schwarz 共Cambridge U. P., Cambridge, 2007兲; String
Theory in a Nutshell, E. Kiritsis 共Princeton U. P., Princeton,
2007兲. These standard string-theory textbooks each include
at least a chapter on black holes in string theory. They are
designed for readers with a working knowledge of quantumfield theory and gauge theories in addition to the usual undergraduate physics material. 共A兲
130. “Resource letter: The nature and status of string
theory,” D. Marolf, Am. J. Phys. 72, 730–741 共2004兲; e-print
arXiv:hep-th/0311044. This Resource Letter contains a more
comprehensive guide to the string theory literature, including
the pre-2004 literature on stringy black holes. 共A兲
131. “TASI lectures on black holes in string theory,” A. W.
Peet, e-print arXiv:hep-th/0008241, in Strings, Branes, and
Gravity: TASI 99: Boulder, Colorado, 31 May–25 June
1999, edited by J. Harvey, S. Kachru, and E. Silverstein
共World Scientific, Singapore, 2001兲, pp. 353–433. Peet’s
thorough introduction to black holes and branes takes the
reader up though a stringy treatment of black-hole entropy
and Hawking radiation. 共A兲
132. “The quantum physics of black holes: Results from
string theory,” S. R. Das and S. D. Mathur, Annu. Rev. Nucl.
Part. Sci. 50, 153–206 共2000兲; e-print arXiv:gr-qc/0105063.
This review takes the reader as directly as possible to the
stringy description of black-hole entropy and Hawking radiation. 共A兲
305
Am. J. Phys., Vol. 77, No. 4, April 2009
133. “Microscopic formulation of black holes in string
theory,” J. R. David, G. Mandal, and S. R. Wadia, Phys. Rep.
369, 549–686 共2002兲; e-print arXiv:hep-th/0203048. The authors review the calculation of black-hole properties from
string theory for the case of the so-called nearly-extreme
D1–D5 black hole. Many important details of the relevant
gauge and conformal field theories are discussed, leading to
black-hole thermodynamics and Hawking radiation. 共A兲
134. “Black holes in string theory,” J. M. Maldacena,
e-print arXiv:hep-th/9607235. Maldacena’s Ph.D. thesis was
not written as an introduction for outsiders, but does contain
detailed treatments of gauge-theory aspects relevant to the
stringy counting of black-hole entropy that are hard to find in
other sources. Other useful aspects of brane and black-hole
physics are also described. 共A兲
135. “The fuzzball proposal for black holes: An elementary review,” S. D. Mathur, Fortsch. Phys. 53, 793–837
共2005兲; e-print arXiv:hep-th/0502050. Mathur reviews a particular set of speculations about black-hole microphysics in
string theory. This proposal is by no means established, but
has attracted significant interest. 共A兲
B. Black-hole microphysics in loop quantum gravity
136. “Interface of general relativity, quantum physics and
statistical mechanics: Some recent developments,” A. Ashtekar, Ann. Phys. 9, 178–198 共2000兲; e-print arXiv:gr-qc/
9910101. Ashtekar provides an introductory overview of
both isolated and dynamical horizons and black-hole entropy
in loop quantum gravity. This work attempts to be as nontechnical as possible, and makes a good first introduction to
the subject. 共I/A兲
137. “Quantum geometry and black-hole entropy,” A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, Phys. Rev. Lett.
80, 904–907 共1998兲; e-print arXiv:gr-qc/9710007. Microstates of black holes are counted in loop quantum gravity.
Their number agrees with the Bekenstein–Hawking entropy
up to issues associated with the so-called Immirzi parameter.
共A兲
138. “Quantum horizons and black-hole entropy: Inclusion
of distortion and rotation,” A. Ashtekar, J. Engle, and C. Van
Den Broeck, Class. Quantum Grav. 22, L27–L34 共2005兲;
e-print arXiv:gr-qc/0412003. The state counting of Ref. 137
is generalized to include distorted and rotating black holes.
共A兲
139. “Isolated and dynamical horizons and their applications,” A. Ashtekar and B. Krishnan, Living Rev. Relativ.
lrr-2004-10,
relativity.livingreviews.org/Articles/lrr-200410/. Ashtekar and Krishnan review the loop quantum gravity
description of black holes and provide a thorough discussion
of background material. 共A兲
C. Other microphysics and black-hole entropy
140. “Black hole thermodynamics from euclidean horizon
constraints,” S. Carlip, Phys. Rev. Lett. 99, 021301–021305
共2007兲; e-print arXiv:gr-qc/0702107. Carlip investigates
whether the Bekenstein–Hawking entropy A / 4ᐉ2p might be
determined by subtle symmetries of black holes, independent
of the detailed microscopic theory of quantum gravity. 共A兲
141. “Black hole entropy as causal links,” D. Dou and R.
D. Sorkin, Found. Phys. 33, 279–296 共2003兲; e-print
arXiv:gr-qc/0302009. Within the causal-set approach to
quantum gravity, Dou and Sorkin show that the number of
E. Gallo and D. Marolf
305
causal links crossing the horizon is proportional to the blackhole area. 共A兲
142. “The entropy of the vacuum outside a horizon,” R. D.
Sorkin, Gen. Rel. Grav., Proceedings of the GR10 Conference, Padova 1983, edited by B. Bertotti, F. de Felice, A.
Pascolini 共Consiglio Nazionale della Ricerche, Roma, 1983兲;
Vol. 2; “A quantum source of entropy for black holes,” L.
Bombelli, R. K. Koul, J. H. Lee, and R. D. Sorkin, Phys.
Rev. D 34, 373–383 共1986兲; “Entropy and area,” M. Srednicki, Phys. Rev. Lett. 71, 666–669 共1993兲; e-print
arXiv:hep-th/9303048. These works introduce the idea that
black-hole entropy might fundamentally describe correlations in the vacuum across the horizon of the black hole. The
corresponding entropy diverges in quantum field theory but
can be made to agree with the Bekenstein–Hawking entropy
A / 4ᐉ2p by imposing a cutoff at the Planck length. 共A兲
143. “On the quantum structure of a black hole,”
G. ’t Hooft, Nucl. Phys. B 256, 727–745 共1985兲. A calculation much like that of Ref. 142 is presented, but from a
different perspective. In this so-called brick-wall model, the
entropy of a black hole is associated with states of quantum
fields outside of, but close to the horizon. 共A兲
144. “Black hole horizon fluctuations,” A. Casher, F. Englert, N. Itzhaki, S. Massar and R. Parentani, Nucl. Phys. B
484, 419–434 共1997兲; e-print arXiv:hep-th/9606106. “How
wrinkled is the surface of a black hole?,” R. D. Sorkin, in
Proceedings of the First Australasian Conference on General Relativity and Gravitation, edited by D. Wiltshire, 共U.
of Adelaide, Adelaide, 1996兲 163–174; e-print arXiv:gr-qc/
9701056. “On the quantum width of a black hole horizon,”
D. Marolf, Springer Proc. Phys. 98, 99–112 共2005兲; e-print
arXiv:hep-th/0312059. We include these works as a counterpoint to Refs. 142 and 143. They argue that a natural cut-off
occurs at a larger length scale, so that the vacuum entanglement entropy is negligible compared with A / 4ᐉ2p. 共A兲
145. “TASI lectures on the holographic principle,” D. Bigatti and L. Susskind, e-print arXiv:hep-th/0002044, in
Strings, Branes, and Gravity: TASI 99: Boulder, Colorado, 31 May–25 June 1999, edited by J. Harvey, S.
Kachru, and E. Silverstein 共World Scientific, Singapore,
2001兲, pp. 883–933. This surprisingly nontechnical work reviews a number of interesting but controversial ideas about
black-hole microphysics ranging from black-hole complementarity to entropy bounds and the holographic principle:
the assertion that the number of states of a region of space is
determined by the area of its boundary. The more established
anti-de Sitter/conformal field theory correspondence is discussed as an example of these ideas. 共I兲
146. “The holographic principle,” R. Bousso, Rev. Mod.
Phys. 74, 825–874 共2002兲; e-print arXiv:hep-th/0203101.
Bousso provides a thorough review of entropy bounds, and
related issues. Applications and examples are presented and
some relevant black-hole physics is reviewed. Much of this
discussion is accessible to nonexperts. 共I兲
147. “Acceleration radiation and generalized second law
of thermodynamics,” W. G. Unruh and R. M. Wald, Phys.
Rev. D 25, 942–958 共1982兲; “Entropy Bounds, Acceleration
Radiation, And The Generalized Second Law,” W. G. Unruh
and R. M. Wald, Phys. Rev. D 27, 2271–2276 共1983兲; “On
the status of highly entropic objects,” D. Marolf and R. D.
Sorkin, Phys. Rev. D 69, 024014–024019 共2004兲; e-print
arXiv:hep-th/0309218. “Notes on spacetime thermodynamics
and the observer-dependence of entropy,” D. Marolf, D.
306
Am. J. Phys., Vol. 77, No. 4, April 2009
Minic, and S. F. Ross, Phys. Rev. D 69, 064006 共2004兲;
e-print arXiv:hep-th/0310022. We include these works as a
counterpoint to some of the arguments in Refs. 145 and 146.
See also Ref. 104. 共A兲
148. “Ultimate physical limits to computation,” S. Lloyd,
Nature 共London兲 406, 1047–1054 共2000兲; e-print
arXiv:quant-ph/9908043; “Black hole entropy and quantum
information,” M. J. Duff and S. Ferrara, Lect. Notes Phys.
755, 93–114 共2008兲; e-print arXiv:hep-th/0612036. These
works explore connections between black-hole physics and
quantum information theory. 共A兲
149. “Thermodynamics of space-time: The Einstein equation of state,” T. Jacobson, Phys. Rev. Lett. 75, 1260–1263
共1995兲; e-print arXiv:gr-qc/9504004. All horizons in general
relativity share the basic features of black-hole thermodynamics. This intriguing paper shows that the usual derivations can be reversed: by assuming that the first law holds for
all horizons in its standard form, one can derive the full
dynamics of Einstein gravity. 共A兲
VIII. CONNECTIONS TO NUCLEAR AND
CONDENSED MATTER PHYSICS
This brief section addresses connections between blackhole physics and that of the at first sight unrelated fields of
nuclear and condensed-matter physics. The resources below
are of two types. First, Ref. 149 reviews the use of established hydrodynamics and condensed matter physics to design analogues of black holes in certain fluid systems and
Bose–Einstein condensates. In contrast, the remaining works
attempt to use black holes and a bit of string theory as mathematical tools to understand quark-gluon plasmas 共Ref. 150兲
and condensed-matter physics 共Ref. 151兲. It will be very interesting to see how these ideas develop in the near future.
150. “Analogue gravity,” C. Barcelo, S. Liberati, and M.
Visser, Living Rev. Relativ. 8, 12 共2005兲; e-print arXiv:gr-qc/
0505065. The authors review the vast literature on analogues
of black holes in fluid and condensed matter systems. Both
classical properties of black holes and Hawking radiation are
discussed. There are interesting connections to Ref. 113. 共A兲
151. “String theory and quantum chromodynamics,” D.
Mateos, Class. Quantum Grav. 24, S713–S739 共2007兲;
e-print arXiv:0709.1523 关hep-th兴. The so-called gauge/
gravity dualities of string theory imply that certain theories
of gravity can be used to calculate properties of what might
appear to be completely unconnected quantum-field theories.
The best-studied example of this is the anti-de Sitter/
conformal field theory correspondence 共AdS/CFT兲. Finite
temperature effects in quantum-field theory are associated
with black holes in the gravitational theory. Mateos reviews
attempts over the last few years to use this correspondence to
understand the physics of quark-gluon plasmas such as those
currently being generated at the relativistic heavy ion collider 共RHIC兲 in Brookhaven, New York. Some background
in gauge theory and string theory is required. 共A兲
152. “Quantum critical transport, duality, and M-theory,”
C. P. Herzog, P. Kovtun, S. Sachdev, and D. T. Son, Phys.
Rev. D 75, 085020 共2007兲; e-print arXiv:hep-th/0701036;
“Theory of the Nernst effect near quantum phase transitions
in condensed matter, and in dyonic black holes,” S. A. Hartnoll, P. K. Kovtun, M. Muller, and S. Sachdev, Phys. Rev. B
76, 144502 共2007兲; e-print arXiv:0706.3215 关cond-mat.strel兴. Strongly coupled quantum effects are believed to play an
important role in certain condensed-matter systems, such as
E. Gallo and D. Marolf
306
high-temperature superconductors. Near a phase transition
these effects should be described by a strongly coupled conformal field theory. As such, they may be amenable to study
via the AdS/CFT correspondence. These papers begin what
will surely be a long process of attempting to do so. 共A兲
ACKNOWLEDGMENTS
The authors thank Amitabh Virmani, Steve Giddings, and
Marta Volonteri for help in locating certain references. They
307
Am. J. Phys., Vol. 77, No. 4, April 2009
also thank Jessica Wirts for assistance in tracking down certain bibliographic information. E.G. is funded by NASA
through a Hubble Fellowship grant from the Space Telescope
Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Incorporated, under
NASA Contract No. NAS5-26555. D.M. was supported in
part by the US National Science Foundation under Grant
No. PHY05-55669, and by funds from the University of
California.
E. Gallo and D. Marolf
307